Convergence Rates for Solutions of Inhomogeneous Ill-posed Problems in Banach Space with Sufficiently Smooth Data

We consider the inhomogeneous, ill-posed Cauchy problem (formula presented) where A is the infinitesimal generator of a holomorphic semigroup of angle θ in Banach space. As in conventional regularization methods, certain auxiliary well-posed problems and their associated C₀ semigroups are applied in order to approximate a known solution u. A key property however, that the semigroups adhere to requisite growth orders, may fail depending on the value of the angle Our results show that an approximation of u may be still be established in such situations as long as the data of the original problem is sufficiently smooth, i.e. in a small enough domain. Our results include well-known examples applied in the approach of quasi-reversibility as well as other types of approximations. The outcomes of the paper may be applied to partial differential equations in L^p spaces, 1 < p < defined by strongly elliptic differential operators.